Recursive Formula

If t₁, t₂, t₃,….,t_n,… is a set of series or a sequence. Then a recursive formula for this sequence will be needed to compute all the previous terms and find the value of t_n.

This formula can also be defined as Arithmetic Sequence Recursive Formula. As you can observe from the sequence itself, it is an arithmetic sequence, which includes the first term followed by other terms and a common difference, d between each term is the number you add or subtract to them.

A recursive function can also be defined for a geometric sequence, where the terms in the sequence have a common factor or common ratio between them. And it can be written as;

Recursive Formula Examples

Example 1:

Let t₁=10 and t_n= 2t_n-1+1

So the series becomes;

t₁=10

t₂=2t₁+1=21

t₃=2t₂+1= 43

And so on…

Example 2: Find the recursive formula which can be defined for the following sequence for n > 1.
65, 50, 35, 20,….
Solution:
Given sequence is 65, 50, 35, 20,….
a₁ = 65
a₂ = 50
a₃ = 35
a₂ – a₁ = 50 – 65 = -15
a₃ – a₂ = 35 – 50 = -15
Thus, a₂ = a₁ – 15
Similarly, a₃ – a₂ = -15
From this we can write the recursive formula as: a_n = a_n-1 – 15

Example 3: Calculate f(9) for the recursive series f(x) = 3. f(x – 2) + 4 which has a seed value of f(3) = 9.

Solution:
Given,
f(3) = 9
f(x) = 3.f(x – 2) + 4
f(9) = 3.f(9-2) + 4
= 3.f(7) + 5
f(7) = 3.f(7-2) + 4
= 3.f(5) + 4
f(5) = 3.f(5-2) + 4
= 3.f(3) + 4
Substituting f(3) = 9,
f(5) = 3(9) + 4 = 27 + 4 = 31
f(7) = 3(31) + 4 = 93 + 4 = 97
f(9) = 3(97) + 4 = 291 + 4 = 295

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